By Taylor series expansion of \(\ln p_{{\textbf {x}}} \left( {\textbf {{\textit {x}}}} \right)\)  about \(\textit {\textbf {x}}\) , (REF ) converges to 0 almost
surely as \({\varepsilon \rightarrow {\bf {0}}}\) . Now, by the Lebesgue dominated convergence Theorem [1]}, to prove (REF ) it suffices only
to show the random variable inside the curly bracket in (REF ) for all \(\varepsilon \)  small enough is bounded by a fixed integrable random variable. Repeating some arguments like those of [2]}, we can write,
\(\begin{array}{l}\left|{{{\left[ {\ln p_{\textbf {x}} \left( \textit {{\textbf {x}} } \right) - \ln p_{{\textbf {x}} } \left( {\textit {{\textbf {x}}} + {\delta }_{\textit {\textbf {x}}} } \right) + {\phi }_{\textbf {x}}^T \left( {\textit {\textbf {x}}} \right){\delta }_{\textit {\textbf {x}}} } \right]} \mathord {\left\bad.{\vphantom{{\left[ {\ln p_{\textbf {x}} \left( {\textbf {x}} \right) - \ln p_{{\textbf {x}} + {\delta }_{\textbf {x}} } \left( {{\textbf {x}} + {\bf {\delta }}_{\textbf {x}} } \right) - {\phi }_{\textbf {x}}^T \left( {\textbf {x}} \right) } \right]} {\left\Vert  {\varepsilon } \right\Vert }}} \right.\hspace{0.0pt}} {\left\Vert  {\varepsilon } \right\Vert }}} \right| \\\,\,\,\,\,\,\mathop < C\left[ {2 + 2^{\max \left( {\alpha - 2,0} \right)} \left( {\left\Vert  {\textit {\textbf {x}}} \right\Vert ^{\alpha - 1} + \left\Vert  {{{\tilde{X} \varepsilon }}+ o\left( \varepsilon \right)} \right\Vert ^{\alpha - 1} } \right)} + \left\Vert  {\textit {\textbf {x}}} \right\Vert ^{\alpha - 1} \right]\left\Vert  {{\tilde{X} }} \right\Vert  \\\,\,\,\,\,\, < C\left[ {2 + 2^{\max \left( {\alpha - 2,0} \right)} \left( {\left\Vert  {\textit {\textbf {x}}} \right\Vert ^{\alpha - 1} + \left\Vert  \varepsilon \right\Vert ^{\alpha - 1} \left\Vert  {{{\left( {{{\tilde{X} \varepsilon }} + o\left( \varepsilon \right)} \right)} \mathord {\left\bad.{\vphantom{{\left( {{{\tilde{X} \varepsilon }} + o\left( {{\delta }_{\gamma } } \right)} \right)} {\left\Vert  {{{\delta }_{\gamma } } } \right\Vert }}} \right.\hspace{0.0pt}} {\left\Vert  \varepsilon \right\Vert }}} \right\Vert ^{\alpha - 1} } \right)} + \left\Vert  {\textit {\textbf {x}}} \right\Vert ^{\alpha - 1} \right]\left\Vert  {{\tilde{X} }} \right\Vert  \\\,\,\,\,\,\, < C\left[ {2 + 2^{\max \left( {\alpha - 2,0} \right)} \left( {\left\Vert  {\textit {\textbf {x}}} \right\Vert ^{\alpha - 1} + \beta \left\Vert  {{{\tilde{X} }}} \right\Vert ^{\alpha - 1} } \right)}+ \left\Vert  {\textit {\textbf {x}}} \right\Vert ^{\alpha - 1} \right]\left\Vert  {{{\tilde{X} }}} \right\Vert \end{array}\) 
